Xie's coherent backward-orbit conjecture

Let $X$ be a quasi-projective variety over $\mathbb C$, let $F:X\to X$ be a finite endomorphism, and let $\{b_i\}_{i\geq 0}$ be points of $X(\mathbb C)$ satisfying

$$f(b_i)=b_{i-1}\quad\text{for all }i\geq 1.$$

Let $V$ be a positively dimensional irreducible subvariety of $X$. Xie's coherent backward-orbit conjecture. If $\{b_i\}_{i\geq 0}\cap V$ is Zariski dense in $V$, then $V$ is periodic under $F$.

This is a special case of Zhang's question about subvarieties meeting a backward orbit densely, analogous to the dynamical Manin–Mumford conjecture. The source presents it as a proposed conjecture for coherent backward orbits.